I am solving the following system of equations (the Streeter-Phelps equation models pollution of a river) by the method of characteristics.
$$b_t + v b_x = -k_1 b,$$ $$D_t + v D_x = k_1 b - k_2 D,$$ $$b(0,x)=b_0(x)\ge 0, c(0,x)=c_0(x)\ge 0, x\in\mathbb{R}, k_1\ne k_2$$
where $b=b(t,x), c=c(t,x), D = c-c^\infty$, where $c^\infty$ is a constant parameter and $v$ is also constant. I am a bit uncertain if my solution is actually correct, especially for the second equation in the system. I would appreciate if someone could please point out any errors.
My solution:
$\frac{dx}{dt}=v$, so that $x=vt+x_0, x_0=x-vt$, and $\frac{db}{dt}=-k_1b$, and so $$b(x,t) = b_0(x) e^{-k_1 t}$$
By the same token, after substituting for $b$ in the second equation, I get
$$D(x,t)=k_1\int\limits b_0(x) e^{-k_1t}dt-k_2 t+const=-b_0(x)e^{-k_1t}-k_2t + h(x) = c(x,t) - c^\infty$$
Also,
$$c(0,x) = -b_0(x) + h(x) = c_0(x),$$
so that $$c(x,t) = -b_0(x)(e^{-k_1t}-1)+c_0(x)-k_2(t)+c^\infty$$
for the first equation $$b_t + v b_x = -k_1 b,$$ $$\frac {dt}{1}=\frac {dx}{v}=\frac {db}{-k_1b}$$ $$\frac {dt}{1}=\frac {dx}{v} \implies vt_x=c_1$$ $$\frac {dt}{1}=\frac {db}{-k_1b}$$ $$t+c_2=-\frac 1 {k_1}\ln b$$ $$f(c_1)=c_2 \implies f(vt-x)=t+\frac 1 {k_1}\ln b$$ $$b(x,t)=f(vt-x)e^{-k_1t}$$ Now plug it in the second equation and solve it.
As pointedout in the comment by @Mattos your integral is not correct. You need to solve the pde with the same method you used for the first one.